Problem 44 - University Calculus: Early Transcendentals

Question

Describe the given set with a single equation or with a pair of equations. The set of points in space that lie 2 units from the point (0,0,1) and, at the same time, 2 units from the point (0,0,-1)

Step-by-Step Solution

Answer

The set is the circle \( x^2 + y^2 = 3 \), lying in the $xy$-plane at $z=0$. It is the intersection of two spheres.

Step 1: Understand the Problem

We need to find the set of all points in space that are equidistant from two given points, $(0, 0, 1)$ and $(0, 0, -1)$. Each point must be 2 units away from both given points.

Step 2: Define the Equations for Distance

Use the formula for the distance between two points in 3D: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \). The distance from a point \((x, y, z)\) to \((0, 0, 1)\) must be 2 units, giving us the equation: \( \sqrt{x^2 + y^2 + (z - 1)^2} = 2 \). Similarly, the distance to \((0, 0, -1)\) must also be 2 units, giving another equation: \( \sqrt{x^2 + y^2 + (z + 1)^2} = 2 \).

Step 3: Express Squared Distances

Square both equations to eliminate the square roots. For the first equation: \( x^2 + y^2 + (z - 1)^2 = 4 \). For the second equation: \( x^2 + y^2 + (z + 1)^2 = 4 \). This converts the square root equations into polynomial form.

Step 4: Simplify the Equations

Expand both equations. The first equation simplifies to \( x^2 + y^2 + z^2 - 2z + 1 = 4 \), and the second to \( x^2 + y^2 + z^2 + 2z + 1 = 4 \).

Step 5: Combine and Simplify Further

Now, simplify and combine the two equations: \( x^2 + y^2 + z^2 - 2z + 1 = 4 \) and \( x^2 + y^2 + z^2 + 2z + 1 = 4 \) can be respectively simplified to \( x^2 + y^2 + z^2 = 3 + 2z \) and \( x^2 + y^2 + z^2 = 3 - 2z \).

Step 6: Analyze the Geometric Shape

Notice that each equation describes a sphere. The first sphere is centered at \((0, 0, 1)\) with radius 2, and the second sphere is centered at \((0, 0, -1)\) with radius 2. The set of points that are 2 units from both spheres at the same time is their intersection, and given the specific constraints, the intersection is a circle in the \(xy\)-plane, specifically the circle \( x^2 + y^2 = 3 \) with \(z=0\).

Key Concepts

Distance Formula in 3D Geometry

In 3D geometry, calculating the distance between two points can be extended from 2D space by using the distance formula. The standard formula used for this calculation is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \). This formula accounts for the differences in each spatial dimension—x, y, and z—thereby allowing us to measure the straight-line distance in three-dimensional space effectively.

To apply this, we consider each coordinate separately:

• \( (x_2 - x_1) \) represents the horizontal distance between the points.
• \( (y_2 - y_1) \) accounts for the vertical distance.
• \( (z_2 - z_1) \) measures the depth or height difference.

The simplicity and precision of the distance formula make it an essential tool for solving problems related to distances in 3D geometry. Understanding this formula is crucial for further exploration into spatial equations, such as equations of spheres and their intersections.

3D Geometry and its Applications

Exploring 3D geometry involves understanding objects in space that have length, width, and height. Unlike 2D geometry, where objects are on a flat plane, 3D geometry introduces depth, allowing for a richer understanding of spatial relationships and object interactions. This type of geometry is fundamental for understanding real-world structures and physics.

In 3D geometry, every point is expressed as a tuple that includes x, y, and z coordinates. This allows for the analysis of shapes like cubes, spheres, and pyramids, each defined by their specific set of coordinates and dimensions. Applications include:

• Design and architectural visualizations
• Graphics and animations in video games
• Engineering simulations

Mastering the concepts of 3D geometry not only helps in solving mathematical problems but also in understanding and creating virtual models and simulations.

Spheres Intersection

The intersection of spheres is an intriguing concept that arises when you consider two or more spheres in space overlapping. In our exercise, we explore two spheres in 3D space, centered at \((0, 0, 1)\) and \((0, 0, -1)\) respectively, each with a radius of 2 units.

The intersection of these spheres involves finding the common points that are equidistant from their centers. For spheres to intersect meaningfully, their combined radial distances must allow an overlap region or common space. In our scenario, solving the equations for these distances as discussed earlier, we find that the spheres intersect in a specific circle within the 3D space. To locate the exact points of intersection, consider the geometric positioning. Solving the equations will often yield a single line or a set of points, which can describe a circle, as in this exercise. The intersection of spheres is a fundamental concept used in mapping regions that can be applied across disciplines such as astronomy and molecular biology.

Equation of a Circle in 3D Space

The equation of a circle in 3D space is derived from understanding intersections in three dimensions. When two spheres in 3D geometry overlap, their intersection typically forms a circle. In the given problem, after solving the equations for distances from two fixed points, we end up with a circle equation in the xy-plane: \( x^2 + y^2 = 3 \). This circle lies at \(z = 0\), indicating the circle is parallel to the xy-plane and represents a flat intersection line in 3D space.

Developing an equation for a circle involves:

• Identifying the plane in which the circle lies.
• Solving the given conditions for equidistance to determine the radius and its center point.
• Employing the reduced equations from spheres to describe the circle in 3D.

By mastering the derivation and interpretation of circles from sphere intersections, students can unlock deeper insights into spatial mathematics, facilitating understanding of complex geometric formations and their applications.